Dispersion of Mobile Robots: A Study of Memory-Time Trade-offs

We introduce a new problem in the domain of mobile robots, which we term dispersion. In this problem, $n$ robots are placed in an $n$ node graph arbitrarily and must coordinate with each other to reach a final configuration such that exactly one robot is at each node. We study this problem through the lenses of minimizing the memory required by each robot and of minimizing the number of rounds required to achieve dispersion. Dispersion is of interest due to its relationship to the problems of scattering on a graph, exploration using mobile robots, and load balancing on a graph. Additionally, dispersion has an immediate real world application due to its relationship to the problem of recharging electric cars, as each car can be considered a robot and recharging stations and the roads connecting them nodes and edges of a graph respectively. Since recharging is a costly affair relative to traveling, we want to distribute these cars amongst the various available recharge points where communication should be limited to car-to-car interactions. We provide lower bounds on both the memory required for robots to achieve dispersion and the minimum running time to achieve dispersion on any type of graph. We then analyze the trade-offs between time and memory for various types of graphs. We provide time optimal and memory optimal algorithms for several types of graphs and show the power of a little memory in terms of running time.